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Contents 1 Task Force Processes 1.1 Developing a Strategy: 2 Vision and Values: 2.1 Vision Statement 3 Mission Statements 3.1 Service 3.2 Scholarship 4 Task Force Objectives (Activies and Issues) 5 General education and entry-level mathematics requirements 5.1 Proficiency requirement 5.2 Advising 5.3 Admission with deficiencies 5.4 Easing placement into remedial courses 5.5 Expansion of readiness testing 5.6 Alternative options for attaining proficiency 6 Mathematics requirements of other majors 6.1 Interdepartmental steering committee 6.1.1 Intellectual connections 6.1.2 Specialized courses 6.1.3 Prerequisites 6.1.4 Data collection 6.2 Statistics courses 6.3 The Role of Mathematics in Advanced Standing in Majors 7 Methods of delivery 7.1 Different Classroom Formats 7.1.1 Class Size: Various Views 7.1.2 Lecture-recitation model 7.1.3 Large lecture, small fall-back section 7.2 Technology 7.3 Quality of Instruction 7.3.1 Training 7.3.2 Highlights on quality of instruction 7.3.3 Rewarding high quality teaching 7.3.4 Communicating Quality Teaching 8 Resources 8.1 Budget Process 8.2 Faculty

[edit] Task Force Processes

Point of Order! It’s time for a “process check” on Task Force goals and how to achieve them. The objectives I listed below were loosely written. Note that these objectives merely describe activities, which we did accomplish, because it’s easy to “consider” something. However, these objectives are not results-oriented. So, what results do we expect from the Task Force? Here is one outsider's over-generalization of what has happened so far: The members of the Math Dept. are asking for more resources. The other (external) Task Force members want to see changes to improve education / instruction. This might be an over-simplification, but it's certainly a good first- or second-order approximation.

At the Science Board of Advisors meeting in December, 2005, Dean Ruiz said that he was looking to “reinvent” the Math Department and that he hoped such a process could be useful for other departments. The Subcommittee on General Education and Entry-level Mathematics Requirements was very successful in making solid recommendations, many of which set new standards. However, even after many meetings, it is not clear what problem the rest of this Task Force is trying to solve. It appears to me that we have been working backwards: unstructured "venting," draft a final report, and try to find consensus recommendations. This has been ineffective. Successful organizations first develop their strategic direction and then later consider the tactical activities needed to support that direction.

[edit] Developing a Strategy:

There are many approaches. I summarize two of the older, proven ones here:

• Michael Porter (Harvard) model:
  • (a) What are the external forces affecting your organization? (List big things beyond your control. Invention of slide rule threatened the slide rule industry.)
    • UA can not be more selective in admission standards for many reasons, fund allocation being an important one.
    • Expected to teach Mathematics to practically all students that come to UA.
    • Expected to have passing rates comparable to those of other foundations courses (i.e. English or Language)
    • Perceived (for better or worse) as contributing to aggravate student attrition
    • Perceived as not putting enough emphasis on "quality teaching"
    • Expected to continue to produce outstanding research at the same time as we meet a larger than normal teaching load.
  • (b) With the resources available, what are your strategic options?
    • One may excel in both teaching and research if not forced to meet the expectation of quantity, i.e. we can serve very well a portion of the students that come. Strategic alliances can take the balance.
    • One may meet quatitative expectations if faculty divides labor along mission lines: Some tenure track faculty dedicated primarily to teach, others dedicated primarily to research. Both need to teach and do research, but in different proportions.
    • One may facilitate the process of achieving "College Mathematics Proficiency" (whatever that means) if allowed to provide instruction at some pre-college levels.
  • (c) Select the best option(s) to pursue.

• C.K. Prahalad / Gary Hamel (U. Michigan – “core competence”) model:
  • (a) Identify the various elements of the “value chain” needed to develop and deliver your product.
    • Raw materials: One may consider being more strict in admitting students that are better prepared in mathematics. The institution must invest in recruiting better qualified students earlier.
    • Supply chain: Expand work with high schools. Perhaps if we form a faculty development plan for HS instructors, the quality of their students will increase.
    • Process: Recruiting better instructors will improve teaching and learning. To recruit better instructors, the University, teh College and the Department must invest in offering more attractive opportunities to prospective teachers.
    • Quality Control: Core Mathematics courses must deliver students that possess adequate command of mathematical ideas that are relevant to their majors. Here teaching must be regarded as a process, like (pardon) manufacturing, where finished products must meet strict quality standards.
  • (b) Identify a small number (no more than two) core competencies that provide a true competitive advantage and in which you must be the best.
    • The Department must excel in teaching of Core undergraduate Mathematics. But the department has now many truly outstanding teachers. One must take full advantage of these teachers and reward them professionally for their teaching.
    • The Department must maintain an active research agenda. The department is full of first-rate research mathematicians and promising young ones.
  • (c) Identify strategic partners who might have core competencies (are best) in the other elements of your value chain.
    • Pima Community College is one partner that has Core competency in the teaching of undergraduate Mathematics.
  • (d) Create strategic alliances with those partners.

Any process for developing strategy must include the following:

• (a) analysis of strengths, weaknesses, opportunities, and threats (SWOT)
  • The Department must engage in a SWOT (Strengths, Weaknesses, Opportunities, Threats) analysis. This Task Force identified some issues that are related to resources and detailed them below.
    • Strengths:
      • A diverse faculty interested all aspects of Mathematics.
      • National recognition for achieving a vertical integration of mathematics instruction from K-8 education through graduate and post-doctoral research training.
      • Inventive faculty who have a demonstrated ability to work with other departments and colleges to design courses that fit the exact needs of their students
      • High quality research program that is equally dedicated to quality mathematics education.
      • Unusually strong cooperative ties with the local community college and local school districts.
      • The strength in Applied Mathematics places it in a strong position to assist the University build in Biology, Optics and other areas.
    • Weaknesses:
      • University and department resources are stretched to the limit.
      • Funding, space, and human resources do not reflect the department’s true size,
      • Size is large enough that corrections would produce hardship for the rest of the university.
      • Growth is restricted but at the same time it is being asked to take on more responsibilities.
      • Not enough statisticians to serve the current needs. Little opportunity to add to this number without sacrificing other parts of its program.
    • Opportunities:
      • Meet the increasing need for specialized mathematics and statistics instruction in other disciplines. It is eager to join the University’s efforts to build a high quality Biology research and instruction program. It is already very strong in theoretical aspects of optics, another growth area in Tucson. The new Statistics IDA is also a perfect for Math place for Math to add quality to another program.
      • Colleges and universities that have made the biggest improvements are those that run the place like a business. If we can recognize the constrains, political, space, money, whatever, that would be good.
    • Threats:
      • Like many other University departments, Math is fast approaching its breaking point. Unfortunately, the central role math plays means that, if it gets there, the results will harm the reputation if the Department, the College of Science and the University. Quick fixes will not be possible, and the remnants of the break will haunt the University for years after.
      • University Budgeting (of departments) isn't rational. Although one has the 22:1 rule (One FTE for 22 students), the legislature tends to be better at deducting when we lose 22 than adding when we gain. Periodically there have to be a major adjustments. The more students we take in the further behind we get, because the tuition isn't sufficient for the entire cost.

• (b) evaluation of the competitive environment
• (c) understanding resource constraints
• (d) plans to overcome obstacles to success
• (e) plan for managing change, especially communications

After a strategy is chosen, tactics are developed to support the strategy. It is a big mistake to first list your favorite tactics (hobbies) and then try to find a strategy that will protect / sustain those hobbies.

Strategy must involve the top of the organization, but others (lower levels and consultants) can help with the analysis.

[edit] Vision and Values:

A vision statement is more important than – and differs from a mission statement. A mission statement describes what you are and what you do. A vision statement states what you want to become. The best vision statements give a compelling, measurable goal with a specific time horizon. (When he became CEO of GE, Jack Welch said that GE had to become #1 or #2 in each of its businesses, or else he would close them. Pitt’s Engineering School states that it will achieve a specific US News ranking within five years.)

Values help describe the aspects of your culture that are considered to be invariant.

Vision and values must be developed from within and must involve the highest levels of the organization. External facilitators might help with the process, but the creators and owners are the same people.

Your vision and values statements are adequate if and only if they can be used as a reference when facing difficult decisions. (Johnson & Johnson pulled tainted Tylenol off the shelves because their values statement clearly put their concern for patients above profits.)

[edit] Vision Statement

The University of Arizona's Department of Mathematics will rank among the best ten (or five?) among public, four-year, PhD-granting higher education institutions in both teaching and scholarship

In regards to teaching, the department should favor quality over quantity, that is, that it should provide the best instruction to as large a number of students as allowed by available resources, but should make no attempt at providing any instruction to all students if the minimum acceptable levels of competency can not be guaranteed. Strategic alliances with partner institutions should pick up the balance.

This is the same strategy they follow in research, namely, that the department excels in some, but not all, mathematical specialties and works to support those areas in which they are already strong.

(Just to have a starting point for discussion)

(Best meaning, ranked top in US New Report? )

[edit] Mission Statements

[edit] Service

The Department of Mathematics provides instruction in Core Mathematics that meets or exceeds the expectations of the University Community.

Students may take their required core Mathematics courses in a timely manner.

(enter a percentage) of students demonstrate command of the core Mathematics concepts that are required by their Major.

The failure rate in Core undergraduate courses should reduce by (enter a fraction) within (enter a number) years

Instruction is rated as very good or excellent by the majority of students and client departments

[edit] Scholarship

The Department of Mathematics aspires to rank among the top 20 public, four-year, PhD-granting, research-intensive institutions in research, scholarship and training of new professionals. (According to phds.org currently the Math Department ranks between 30 and 42 depending on weight given to different criteria. Aspiring to be in the top 20 seems like a reasonable goal.)

[edit] Task Force Objectives (Activies and Issues)

• Conduct a sweeping look at the relationship of the Math Department to:
  • Other departments' requirements
  • Community college and high school math preparation in Arizona
  • Recruitment and retention of UA students
  • UA's future in serving an expanding college-seeking population and in outreach programs
• Consider how core math courses might be revised and reorganized to better serve:
  • University-wide General Education Program
  • Special requirements of math-intensive majors and certain other departments
• Consider the following issues:
  • Math preparation of incoming students (relative to our chosen standards)
  • Entry-level math requirements
  • Math placement policies (aims, prerequisite enforcement, processes)
  • Is college-level math needed by all students?
  • Different entry-level math requirements for different student groups (depending on their plans)
  • Math courses designed to serve particular majors
  • Degree of distinction between basic and higher math, with the latter geared to math-based majors
  • Articulation between math at UA and at community colleges
  • Informed advising across UA about math and math options
  • Class sizes and staffing types for different levels / types of math
  • Use of alternative strategies within process of curricular renewal
  • Sustainability fo current course offerings in Math and math-related areas
  • Math and instructional technology
  • Preparation of public school math teachers
  • Support for students in math beyond courses (how much and what kinds?)
  • Coherence of the UA student experience and Math's role in it (from student and faculty perspectives)

[edit] General education and entry-level mathematics requirements

[edit] Proficiency requirement

The General Education/Foundations Math requirement at the UA should become more clearly a “Proficiency” one (as in the Second Language requirement), though the present structure of G, M, and S “strands” should remain for students who cannot meet this Proficiency initially or who need levels of math beyond this basic one. We believe that what constitutes Proficiency should be determined mainly by the Math Department itself and then connected with a list of options (from AP and CLEP exams, say, to a particular course or a certain score on a the SAT, ACT, or MRT) by which that Proficiency can be demonstrated by entering students or at any time prior to a student attaining 56 units of credit (with transfer students having one semester to meet this standard, as is the case now, if they have not done so at the previous institution). If a student demonstrates that Proficiency and does not need or want any more math for a major, a minor, or other interests, there should no longer be the requirement of three course units, even if there are now ways of testing out of these too. The Math Department may in that way be relieved of having to serve some students who need no math beyond a basic college-entry level, and yet there will still be an academic standard and skill level in math that we will ask all UA undergraduates to have attained (and to have verified) by the time they become juniors. This change should, of course, be widely publicized throughout Arizona’s secondary schools and community colleges so that students there know what level of proficiency is expected and have more incentive to attain that level before or around the start of their first year at the University.

[edit] Advising

There should be improved communications to, and more systematic advising of, secondary-school students to encourage the taking of algebra-based math in the senior year of high school. The success rate in college for students who have actively studied math as seniors is much higher than for students who have not (and “taken a year off” from it after doing four years or less from the 8th to 11th grades). In recommending this, to be sure, we recognize that some high schools are now having difficulty providing enough math for seniors because of pressures to get all their students to pass the AIMS test in math earlier. Where there are such cases, we suggest the enrollment of high school seniors in math as it is offered by community colleges, preferably at the level of College Algebra or (if necessary) the course immediately prior to it. That may mean some changes in the dual enrollment process with community colleges, but Pima Community College for one (hereafter PCC) has promised to work on making those changes, for which we are grateful. If a student in that way is able to pass the equivalent of Math 110 or even 105 in a manner that is transferable for University credit, of course, that student should be seen as having satisfied the UA Math Proficiency requirement.

[edit] Admission with deficiencies

Those students admitted with deficiencies in math (such as three instead of four years) must be sent a clear, though supportive, message as early as possible that explains what must be done – and the options available – to correct the deficiency and attain the Proficiency requirement. We believe that this message can and should be communicated in a manner supportive enough to keep many students from simply declining to enroll. But there should still be a clear itinerary communicated early to students with math deficiencies that incentivizes, and points to avenues for, the rapid make-up of the deficiency and the attainment of at least Proficiency by the time such students reach 56 units of credit.

[edit] Easing placement into remedial courses

The enrollment of entering students into pre-UA level math, depending on their MRT placement, should become more seamless at the UA than it is now. First, working with PCC – and not at the UA’s expense – we should increase the number of PCC Math classes, at their 122 level and below (the levels prior to our Math 105 and 110), that are taught on the UA campus (with PCC instructors paid by PCC). For those students who cannot meet the Proficiency requirement and/or place below our entry-level Math classes on the MRT, they should be scheduled, seamlessly for them, into one of the UA sections of Pima Math 122 or below, depending on their placement. These students should be told at the earliest possible point (perhaps including the one in # 3 above) that they must pay the extra cost at Pima’s rates, but we should be able to bill them for that ourselves once they’re here, and they should be able to gain the needed admission to PCC (perhaps online) without traveling there. [This last part of this recommendation, in fact, can take place as soon as possible.] To avoid conflict with the unit totals required for full-time status at the UA and the financial issues that can result, the UA should grant elective (not Math nor Foundations nor Gen. Ed.) credit for Pima Math 122, though not for any level below that. To this end, Math 122 should show up on the UA Course Equivalency Guide as a general elective. That credit will not gain students the completion of the Proficiency requirement by itself, but it will set them up for its completion in a more advanced course or by examination. [This recommendation should go forward only as part of this Task Force’s final report.] At all times, the UA must maintain clarity about the distinction between Proficiency, on the one hand, and placement in math courses, on the other.

[edit] Expansion of readiness testing

For the implementation of # 4 above, Math readiness testing should be expanded or adjusted to calibrate where students should be placed below Math 105 or 110 and even below Pima Math 122. If we are going to send students to these pre-college levels on our campus, we need to be able to place them more precisely (in, say, Pima Math 92 or 86) than we can now.

[edit] Alternative options for attaining proficiency

For those students whose majors place them in what is now the “ G” strand and who cannot initially attain Proficiency by any of the other avenues for establishing that, PHIL 110 and LING 187 should be options for attaining Proficiency for any student who places at the Pima Math 122 level or above (for PHIL 110) or UA Math 110 or above (for LING 187). In other words, if a student places at the Math 122 level or above – but not at the Proficiency level – on the MRT, he or she may take Philosophy 110 for 3 units and attain the Math Proficiency requirement by achieving a passing grade in that course. For a student to use LING 187 to fulfill the Proficiency requirement by passing that course, he or she must gain entrance to it by placing in the MRT at least the UA Math 110 level.

[edit] Mathematics requirements of other majors

[edit] Interdepartmental steering committee

We propose an interdepartmental steering committee that would

• Foster intellectual connections between mathematics and other disciplines
• Consider requests for specialized courses from other departments and coordinate the needs of other departments with existing mathematics courses.
• Provide a standard way that instructors of follow-on courses can raise issues with the committee---a sort of bug report form to help iron out glitches.
• Have access to a standard set of data each year to see how prerequisites are working.

[edit] Intellectual connections

Since one can only teach well what one knows well, it is important that interdisciplinary curricular development involve faculty from different departments who also share research collaborations, at least at some level. Many mathematics faculty would like to be involved in developing courses that fit the needs of other departments in the university, and that it would be useful if some mechanism were put in place to facilitate such activities. For instance, time is something we all lack and one could imagine having a "faculty in residence program" in which mathematics faculty would get a reduced teaching load in exchange for spending time assessing the mathematical needs of students in a particular department, and being involved in the development of the corresponding courses. This would also have the advantage of freeing some of our time to discuss research with faculty in other departments and explore possible collaborative projects, or further existing ones. None of these cross-disciplinary courses should be cast in stone and there should also be a mechanism to facilitate their evolution, based on the needs of the client department.

At the same time we must encourage faculty in client departments to use mathematics in their classes, especially if mathematics is a pre-requisite for their courses. The working group report on on the biological sciences indicated that faculty in the biological sciences use mathematics in their research but not in the classes they teach. Students will make the connection between mathematics and their discipline when they see mathematics used as a tool in their major courses. The steering committee could foster the formation of advisory groups to facilitate this. These would be ad hoc and limited in scope. If students don't see mathematics as a critical tool in their major, they will continue to complain that mathematics is only a requisite and will continue to find it a burden. On the other hand, when faculty begin to use mathematics as a regular tool, students will feel they really need to get it and will apply themselves to study it better. A model is the modules being developed to add quantitative content to the entry-level biology course. However, we must find ways of urging that faculty actually use these modules.

[edit] Specialized courses

The steering committee would provide guidance on the choice between the following two models.

On the one hand, there is a clear need for special mathematics courses in some areas, such as the life science students. In Bio 2010 the National Research Council recommended we train two kinds of students, research biologists and quantitative biologists. Students enrolled in the quantitative track could take courses like 250 A&B (calculus + differential equations), followed by 362 (Probability, calculus based), 363 (this would be a calculus-based statistics course; it does not exist yet, but we talked about it in the Department) and modeling (485 or 380). We would have to make sure all of these courses contain enough biology-based examples and assignments. For the research biologists, we would have to create a special course that covers a selection of topics from the above courses, at a less in-depth level.

On the other hand, there are some concerns about specialized courses. For example, some faculty in Social Sciences reported that they would like their students to mix with with engineers and physical science students, because it pushes their students a little harder. Another concern is that mathematics courses that are listed specifically as "Mathematics for XXX," where XXX is a subject which traditionally had fewer math requirements, will appear to be watered down and less mathematical, even if this is not the case.

The steering committee could provide a condition for acceptance of "Mathematics for XXX" courses: that faculty in XXX will use the said mathematics contents in subsequent courses and graduates are expected to use said Math in their professional lives.

[edit] Prerequisites

Mathematics prerequisites should be taken seriously by the students, the client department and the mathematics department. Client departments should confirm that prerequisite material will really be used, the mathematics department should confirm that it will provide the needed material. In return, students should be willing to work for more than a minimal passing grade and take prerequisites in a timely manner.

Policy recommendations that flow from these principles could include enforcing the prerequisite at the level of a C or higher, and require that the prerequisite be taken less than three years before the follow-on course. Another possibility is prerequisite exams at the beginning of the follow-on courses. Note that automatic enforcement has to be done carefully to look for mathematics courses more advanced than the prerequisite. We note that ECE opts out of online reg because it does not enforce prerequisites.

Notice that there could be limitations on electronic prerequisite enforcement. Some requirements (e.g., ability to lift 45lbs, a prerequisite for nursing) cannot be enforced this way.

[edit] Data collection

The steering committee would design and supervise a system of annual reporting on

• Success rates in mathematics classes, percentages of As, Bs, and Cs
• Prequisite compliance on follow-on classes
• In certain cases flagged for special consideratin, grades in follow-on courses as a function of prerequisite grade and instructor

With the assistance of Institutional Research, Enrollment Research, and CCIT, a Decision Support System should be put in place to make sure the process of data collection and organization is efficient and complete. The system should automate the production of relevant reports and analyses. The system should include qualitative as well as quantitative evaluations to produce a more robust profile of the state of the system. The Steering Committee should have an advisory, but not executive, role in the evaluation process. The Math Department's leadership should retain executive privilege and responsibility.

[edit] Statistics courses

Currently the teaching of statistics is dispersed in many departments.

For example, the College of Engineering (COE) has 4 introductory probability/statistics classes:

• SIE 305 (fall 110 students - spring 120 students - summer 35 students)
• CE 310 (spring 63 students)
• AME 474 (fall 35 students)
• MNE 402 (fall no data as instructor was on sabbatical)

The courses overlap by at least 9 weeks of material, and are about 2/3 probability and 1/2 statistics. A common argument against combining them is the need for students to see examples in their particular major, although there is no data to support this argument. A counter-argument is that probability and statistcs is taught on many campuses as a generic vanilla course, and it should be possible for a good instructor to weave examples from many fields into the course. It requires getting the examples and as we have so many instructors in this class, it would seem that the examples exist. If the university decided to implement a generic probability and statistics class, some engineering programs would participate and others would not. Engineering would contribute to such a course by providing both examples and contributing to the teaching load.

On the question of technology, the engineering classes use a mix of Excel and special packages such as Minitab. Standardizing the technology choice would be a good idea as then instructors and students would know what was expected and what would be provided.

[edit] The Role of Mathematics in Advanced Standing in Majors

The last few years the University has seen a growing trend toward secondary admissions standards for specific majors or entire colleges. Invariably math plays a major part in these standards. Often one or more math courses must be completed for advanced standing and an overall GPA at a certain level is required. Recently, however, specific grades in mathematics courses are being required before a student is admitted to a major. It was once popular to say, “Mathematics should be a pump, not a filter.” Right now at the University of Arizona, mathematics is far too often only a filter. It is hard to estimate how expensive it is to use mathematics this way. We know for a fact that many students take math courses multiple times: not just 2 or 3 times and not just because they failed! The cost of mathematics instruction is never part of the discussion when these upper division requirements are set.

There is, however, a good reason for requiring specific grades in mathematics for specific majors: a certain level of mathematics competency is essential in a growing number of disciplines. This is clear in science, engineering, business, agriculture and architecture, and, although there is less agreement on this, in the social and behavioral sciences. Even the humanities and fine arts have a place for people with mathematics training, although many students in those areas are afraid of the subject.

Unfortunately, mathematics will always be an obstacle that prevents some students from finishing the first program of study that they attempt. We need to accept this and help these students move on.

To that end, we should establish a mechanism that discourages students from enrolling in a math class more than 2 times. It should discourage, not prevent; there will always be those that can overcome difficulty through persistence. But we should encourage those people who do not have that level of persistence to make a change in direction that better fits their talents.

We earlier mentioned that if prerequisites are to be enforced, then the mathematics in the prerequisite courses must be used in the follow-on courses. The same applies to upper division standing requirements. If mathematics is to be a requirement for upper division standing in some program, upper division courses in the program must use that mathematics. Further, instructors of these courses must be more willing to grade students on their mastery of that prerequisite mathematics. Artificial grade requirements in mathematics courses to enter majors where mathematics ability is never graded again are counterproductive at every level. The University should have in place policies that are enforced on advanced standing criteria that assure that all entrance requirements are actually used, required and graded honestly in later courses where advanced standing is required.

[edit] Methods of delivery

The Task Force notes and supports the Mathematics Department’s fundamental pedagogical philosophy: that students learn mathematics best through active learning techniques, such as discussion among students, experimentation (individually or in groups,) peer-to-peer teaching and other forms of direct student engagement over traditional forms of classroom environments (mostly listening and taking notes). The effective use of technology (PowerPoint Presentations, interactive web sites, graphing calculators, etc.) is also well received by students.

Active learning works best when students are intrinsically motivated to work. Students who are intrinsically motivated to study have a personal stake in the success of the process. They find learning the material valuable and personally relevant. Many students do not have that kind of feeling towards mathematics, at least in the foundational and beginning courses. This places an additional burden on the instructor who needs to plan the lessons carefully to introduce motivators.

[edit] Different Classroom Formats

It is important to keep in mind that because the mathematics department is so large, efficiency measures that might be tempting in another department have a very small relative effect on the mathematics department.

For example, changing a few higher courses to large lectures (100-300 students) saves some regular faculty resources, but does not really increase the total amount of teaching mathematics can do by that much. The department has already made this change in Math 254 (Differential Equations), and is talking about it in Math 322 (Math for Engineers). This leads to a gain of 4 or 5 faculty teaching slots, but only 2 or 3 sections, as GTA teaching is used differently. Even the gain of 8-10 sections that might result from a more draconian model for instructor and GTA use is only a drop in the mathematics teaching bucket.

Therefore, to make a significant difference in resources, change needs to take place in the larger courses, particularly algebra and calculus. Business Mathematics, though relatively small, is worth looking at also because it is an expensive to teach. However, it is not at all clear the the University has the lecture room space to accommodate a format change in any of theses courses.

Furthermore, a transition period will be more expensive and require more resources than both the current system and the final one. The size of the courses is such that without careful planning the transition could get very out of hand. Even with planning, the University needs to have back up reserves just in case things do not go as smoothly as expected. One bad section means 100 underprepared students spreading problems into other courses for the next three years.

[edit] Class Size: Various Views

Opinion on class size varies considerably within the Task Force.

Some make the following case for large lectures. First, there are economies of scale. Secondly, it is easier to set and improve standards, to enforce consistency, and to maintain quality control over the educaton process. Also, sections that are interchangeable have advantages with respect to subsitution of instructors and students can attend alternate sections (when necessary). Thirdly—and perhaps most importantly—large lectures cause students to become more self-reliant. After graduation, students will no longer have the benefit of small classes and interactive learning. The most productive members of society must be able to take the initiative to learn on their own from whatever means are available–for example, reading text books, reference manuals or computer help screens. It is the responsibility of the university to make sure that, by the time a student reaches the university level, the onus for learning is on the student, not on the faculty. It is up to educators to set the standards–the expectations–for students to learn the subject matter. We can help the students develop the learning techniques that will work best for them for the rest of their lives. The university provides exposure to quality lectures and other delivery media, and we will provide other support for the learning process (recitations, tutoring, libraries, etc.)–not just to make sure that the subject matter is absorbed, but also so that the student can learn by him/herself.

Others, including many veterans of the department's last foray into large lectures, believe that small classes have several advantages over large lectures, and make the following argument. They are preferable because they foster a close interaction with students. They tend to work well, especially, in courses where students are not confident in their mathematical skills. The reality is that many entering students are not confident in their mathematics skills, do not have good study habits, and are not mathematically mature to learn on their own. Experience suggests that we need to help them in this transition to become independent learners. It takes a year or so for students to develop his or her confidence and knowledge base to where they can take more of the responsibilty for their own learning. Small classes provide the instructor a better opportunity to keep track of the progress of students and intervene at the first sign of trouble. This in turn tends to make students more likely to keep up with their course work and more accountable to their instructor. This is especially the case for courses like college algebra and calculus I where students are not mathematically mature and need the extra attention. Another advantage to small classes is that most instructors are comfortable in that environment.

Yet another point of view considers the dichotomy between spoon-feeding and self-reliance represented in the arguments above to be a false dichotomy. There are many ways to organize the teaching of lower level mathematics. The success of any one of these depends more on the effort put into the organization than the format used. The effort necessary for mathematics instruction to be successful must extend far past the Mathematics Department alone. There must be an institutional commitment to supply the proper classroom space, the academic assistance, the bureaucratic support, the teaching and support staff and the financial means necessary for the system used to succeed. Any reformulation of the way Mathematics is taught at Arizona, or anywhere, that starts out primarily as a means to reduce more than one of these essential components is doomed to failure. It may be possible to concentrate on reducing the Mathematics department’s dependence on one of these, but only if the others can be increased at the same time if the result is to be quality mathematics instruction.

In general, large lectures tend to work well with more advanced courses, when faculty specialize in such assignments, and are assigned to teach the courses on a long term basis. There must be a core of faculty who agree to make a specific large lecture course a significant component of their teaching responsibility. In Sunday's Arizona Daily Star article 5/7/06 on creating named chairs in departments at the University, Gerald Swenson, Thomas R. Brown Chair in Economics, a professor with more than 20 teaching awards, said the following about large lectures: To take on these large classes, you've got to devote a lot of time. Somebody who decides to take this on has to decide this is their focus - teaching. The department's recent experience with large lectures in Math 254 (ordinary differential equations) is successful because a small group of faculty have taken this assignment and the mathematical maturity of students in the class. Furthermore, large lectures should be designed to foster active learning. This can be achieved with two tools: technology and peer assistants. With these two, one can introduce on-line materials, concept tests and challenge problems using class-participation technology such as clickers, just-in-time teaching and other techniques that do not require technology, such as a problem of the week. This would require putting the best teachers in these classes, without rewarding weaker teachers with more desirable class assignments. Thus teachers of large lecture would need some teaching unit release. The department's instructors can learn to teach in large lectures, but this would require resources both for the training and to allow the learning. The transition will take much longer that most people expect and dealing with the problems along the way will be costly.

This is also tied in with the resource issue. There is also strong evidence that, at the moment, the University is not set up to implement a lecture-recitation format in lower level mathematics instruction. The greatest immediate problem will be proper lecture hall space. The best lecture rooms in the 150-300 seat range are already solidly booked. Further, these facilities were built with specific programs in mind, and those programs have first claim on their use (and in some cases practical exclusive use.) It is important to note that when Math 110, 112, and 129 have their common exams at the same time, they occupy basically every large lecture hall on campus we are permitted to use.

Another resource issue is the availability of quality instructors for the small-class format. Business math is a case in point. Being a new course, a new type of course, this would be a good place to experiment. It would require a whole new design to move it into a large lecture format, but this could well be possible. While it is our most expensive course, that comes in technology, class size, and instructor preparation time. It is large enough to have some final impact on required resources. At the same time, the transition period would be more expensive than a more traditional course. In the end, however, the question would come down to the availability of technology classrooms (both big and small) that fit a sound lesson plan. There will be tremendous pressure to plan a course based on limited university technology classrooms rather than on the real needs of the students.

The history of the Mathematics Department is such that many of faculty members that have been here long enough have a very strong aversion to the large lecture format at Arizona. In the 70's and early 80's the department experimented with large lectures of size 300-600. It was a total disaster with attrition rates as high as 50%. There was dissatisfaction on the part of students, math faculty, and frustration from advisors and faculty in other departments. The courses were so unpopular that the department had to resort to a schedule where every faculty member had to take a turn. These and other frustrations led to a University task force appointed by Provost Hasselmo in 1983. The recommendations of that report led to the current scheme of small lectures and significant improvements in the curriculum. Granted that the bad experience is over 20 years old, but it was bad enough that it is unforgettable to those who experienced it. It took the Department years to overcome the ill will and the horrible reputation it deservedly received during that time when it neglected entry level mathematics. The Mathematics faculty must be forgiven for forcefully fighting any plan that even might point us back in that direction.

Mathematics is willing to try its hand at large lecture formats when it can, courses with 100 to 300 students. It does appear as though this does save on faculty level instructors, and perhaps on instructors in general. However, it has made it more difficult for students to schedule the classes, and client Departments have complained about the resulting lack of opportunity to schedule the course. Variability in the course in any one semester has certainly been reduced, in content, standards and grades. Variability semester to semester, however, is a new problem the Department is only just trying to deal with. Course grades have seen a sharp rise under the large lecture format, possibly due to better learning from better instruction, but other explanations are available.

Mathematics is much more reluctant to try to change the format of out larger courses. First, these courses are so large that few outside Math understand how massive a project that would be. Second, such a move would be expensive and take time to plan and execute, and it is not clear that the University could really muster the effort it would take to carry it out. And finally, there is the problem of bad past experience. Many people who would have to play a major role in making the change have an honest fear that promises made at the beginning will not be kept in the end. To get those people involved, the University would need to show its support for such a change with irrevocable tangible resources. If the main motivation for making the change is to reduce costs, it will be hard to find the leadership necessary to carry it out well. The Department would rather not do something at all, than to do it poorly.

[edit] Lecture-recitation model

In mathematics it is not possible to teach well in large lectures without a major recitation component. It is important to resolve the question of how much university credit is given for a recitation hour, and how funding to the mathematics department is calculated for classroom hours as opposed to university credits.

The lecture-recitation format is less successful in Mathematics when the recitation instructor's role is reduced to problem-solving aid. Thus, the format works best when the recitation teachers attend the lectures as part of their assigned duties and there are regular planning meetings with the lead faculty lecturer. This keeps them informed about not only about course content, but also about the method and language of the instruction. This is the potential hidden cost in large-lecture format. If recitation instructors, typically GTA's, are required to attend the lectures and planning meetings that time must be credited against their teaching load. Also, scheduling recitation instructors into both lectures and recitation sections could result in a overall loss of efficiency depending on the final weekly schedule of the class.

Any move toward this format in any course should be only start after a period of solid planning. The intermediate and eventual success of the change will depend on a proper allocation of resources. Assignment of credit to lecturers, recitation leaders, and peer tutors is critical. Availability of rooms will determine the timing of lecture and recitation sections. This timing will be important in estimating the number of lectures needed to accommodate student schedules and in estimating the distribution of instructor resources. If any new format is to be sustainable, we must move cautiously and carefully at each step along the way.

[edit] Large lecture, small fall-back section

Most TA's can do recitation, but teaching fall-back sections requires extra experience and training. Again it is a mistake to believe that fall-back sections can work without creating close ties between the lecturers and the fall-back instructors: the closer the ties, the greater the cost, and the lower the savings over small sections. This can be done in a way that increases the quality of instruction given to most students, and so it could well be worth the cost of the change. If TA's are reduced to doing recitations only then they are not provided with the opportunity to develop their teaching skills by being 100% responsible for the instruction and management of courses while under the supervision of an experienced faculty mentor.

[edit] Technology

Mathematics courses, especially Core Undergraduate, may (orig. should) incorporate the use technology to help all students progress in their learning (This from a position document by MAA.) Technology (should) may be used:

• Effectively as a tool for solving problems;
• As an aid to understanding mathematical ideas.
• When appropriate, as a tool in the application of Mathematics to a specific field.
• As a tool to facilitate grading and other administrative functions of the class.

The use of technology must be accompanied by a thorough and substantial revision of the curriculum. Introducing 21st century math enabling technology into mid 20th century curriculum (in a 2006 funding environment) requires change and compromise. If this is the direction The Department wants to take, then it must work to achieve its clients' consensus. If the Department is going to teach students to use certain kinds of technology tools, client departments should prepare to see their students use these tools to solve some types of problems. Achieving compromise and consensus to identify, accept and adopt changes requires a large amount of work and a certain amount of time.

Issues in the adoption of classroom technology:

• Standards. It is often difficult to make recommendations because people tend to use their favorite software and may not welcome suggestions to use different products. However, for Core Undergraduate Math courses, selection of technology must be coordinated with clients.
• Access. Technology should be affordable to students who own a computer and readily available on campus to students who do not own a computer.
• Training issues. All Instructors must achieve an acceptable level of mastery in the technology used prior to use this technology.
• Planning issues. For all courses where a form technology is used for the first time, a brief tutorial in the use of the technology should be included in the curriculum. There will be students with little or no experience with the technology in question and must learn enough about the program to do the assignments.
• Consistent use. If technology is used in a given course, all sections of the course must assign same or equivalent projects that use the technology
• Relevance. Assignments must have an explicit learning objective related to an aspect of the technology being used
• Student support. Adequate technical support in the use of technology must be available to students. This support should not be provided by the Math faculty.
• Refresh. Once a form of technology has been adopted, long-term plans for refresh and upgrading must be put in place. These plans should include a permanent budget allocation.

[From Uribe - Please consult the discussion for some language that I took off this section, but the may be reused if found appropiate]

[edit] Quality of Instruction

The Department should guarantee that students will find an instructor with the ability, temperament and knowledge to facilitate learning.

An important, but difficult aspect of service teaching is quality control. It is hard because it is often mistaken for violation of academic freedom. However, the Department must ensure that students receiving adequate credit for their Core Mathematics courses demonstrate a consistent command of the core Mathematics concepts that their major-specific courses require them to have.

The department already does much in this regard through common contents and common finals but it must continue to work in this area, especially in regards to teaching. The department must ensure that all instructors of a given Core undergraduate course have common pedagogy, methods of delivery and teaching tools.

There is always an inherent risk of hiring poor-quality instructors. It is normal in any line of business so this task force should make no promises about avoiding the risk. However, The Department must take every possible measure to minimize the risk of hiring poor-quality instructors.

The Department has an internal evaluation system that removes poor-quality instructors from teaching. The Department must expand its internal evaluation system to include (if it does not have yet) a clear statement of expectations and a thorough assessment protocol. It must put this system in place, make it clear to every new hire (including TAs) and apply it strictly, especially when performance falls below expectations.

[edit] Training

Currently, The Department has in place an extensive program to train Teaching Assistants. New TAs:

• have a one-week orientation,
• must take a one-credit, 2-semester course, and
• have faculty supervisors who visit their classes twice a semester.

This Task Force recommends the expansion of this training program to all new instructors (faculty and lecturers included). The Department should extend this program into a reasonable, well-articulated and feasible internal strategy on faculty development. Within it, instructors of (at least) core undergraduate courses should be required to undertake regularly (and be rewarded for undertaking) training or other professional activities that enhance their ability to teach effectively.

[edit] Highlights on quality of instruction

It is important to have a clear definition of high quality instruction. The following are just a few attributes of high quality teaching:

• It is not about entertaining. However, a high-energy classroom environment does help hold the students’ attention.

• It is engaging and inspirational. Visible instructor enthusiasm and love for the subject helps, especially in foundational and general education courses.

• It is about knowledge. Instructors should demonstrate thorough command of the subject.

• It is not about high grades. Higher levels of performance are a consequence of the good learning derived from the teaching.

• It is not about low grades. On the contrary, it should make the material more accessible, and its learning more enjoyable.

• It is about student learning. Students should feel empowered to tackle successfully problems that require the acquired knowledge. This should be demonstrated in class primarily, where the instructor can and must provide immediate feedback, as well as through assigned work, tests and research projects.

• It is not about accent. However, clarity of exposition is critical for good understanding of the material. Efforts should continue to develop presentation and exposition skills in all teaching staff

• It is about presentation. Clarity is not restricted to verbal recitation; materials should be developed carefully and presented in different ways to reach to as many different types learners as possible. Technology may and should be used, when appropriate, to complement classroom exposition.

• It is about self-assessment. Instructors should ask constantly the following three questions: What do I expect from the students? How, and how well do they learn? How can this learning be facilitated?

• It is about knowing one’s own students, where they come, what the feel, how they learn

• It is about valuing interaction. Instructors should request and provide feedback constantly.

• It is about more mentoring and less lecturing. Instructors should model the skills and behaviors the students are expected to learn.--150.135.169.243 15:52, 12 May 2006 (EDT)

Very few math instructors understand points about quality teaching instinctively; most need to learn them over time, and there are a few who never fully accept them no matter their experience.

Developing high quality teachers (not just instructors) is a long process. The department has the tremendous opportunity to incorporate this development process in the formation of its graduate students (themselves faculty in training) as well as the development of its junior faculty. The department is one of only a few units on campus whose service mission is at least as important as its academic one.--150.135.169.243 15:52, 12 May 2006 (EDT)

[edit] Rewarding high quality teaching

The Department must retain its best instructors and help its poorer instructors move on quickly. The Department must work with instructors and give them the time necessary to grow. This is especially true for graduate students, post-docs but also applies to regular faculty. The faculty development plan should address specific needs for all these groups.

One way to contribute to increase quality of instruction is to reduce the need to use GTA’s as classroom instructors. If the Department could assign more GTA’s to teaching duties other than classroom instruction, it would retain more flexibility and be better able to deal with unexpected problems later in the semester.

[edit] Communicating Quality Teaching

The Department needs to develop a deliberate strategy to communicate continuously with the community on its overall academic work, but in particular its efforts and successes on improving quality of instruction. Efforts to maintain links with clients are "not often cited as an important part of the Department's mission, and therefore continued support for these efforts is difficult to maintain."

Oftentimes perception is everything. There seems to exist a (perhaps unfounded) perception that the Department's instructors are poor teachers. In this case, no matter what The Department does, their instructors will continue to be perceived as poor performers, particularly if this perception places the blame for student failure on the Department. Although the Departments has always acted promptly on legitimate complaints, The Department needs to fight this misperception. It needs to publicize its leadership in teaching and learning, its high marks in course evaluations, etc. especially to students and advisors.

[From Uribe -Resources for Instructional Design was moved to dioscusion]

[edit] Resources

[edit] Budget Process

The Department must advocate for a more rational budgeting. Perhaps we should ask ourselves which is greater: the incremental cost of including a student in a course, or the amount of tuition that student pays? and derive strategic decisions from the answers to this and other critical resource questions

The University needs to devise a better mechanism for funding freshman level courses. The funding for freshman courses should be stabilized so that the hiring of contract instructors can be completed early enough to insure quality.

The University should minimize the risk of “last minute” hiring through better early estimates of the number of students Math will see each semester.

[edit] Faculty

In recent years, Calculus has been totally reformed, Business Math has changed, College algebra and Trigonometry are undergoing changes. Thus, the type of faculty we need to hire nowadays is very different. The Department needs to invest in higher quality faculty. It needs to look at long-term teaching positions (lecturer, even teaching faculty, both requiring Ph. D.)whose primary focus is collegiate education; it needs to expand the teaching postdoctoral program (which has lost 1/3 because of budget cutbacks); it must involve tenure track faculty in curriculum innovation and reward them professionally for this work. It also needs to build or expand a cadre of research faculty with a strong interest in education, in particular Mathematics Teaching and Learning. Moreover, Faculty workload agreements must be reached on a rational basis, allowing for different ratios (among teaching, research and service) with evaluation and rewards set accordingly.

Salaries should better reflect the current market for quality math teachers to the point where the University can compete on a national level. The number of permanent Lecture salary lines in the Department should increase to reflect the University's need for people with specialized teaching skills.

==Management and Planning. Over the past several years, we have seen a deterioration of Business Math (MATH 115a and 115b). One possible explanation for this is the manner in which the courses are staffed. In particular, presently no tenure-track faculty teach either of these courses. The bulk of the teaching is done by adjuncts who are on a one-year appointment. Furthermore, the Mathematics Department funds these positions from "temporary funds"--the availability of which is uncertain until the summer before classes start. One possible way to improve matters and hopefully better integrate the pedagogical needs of the university with the intellectual capital of the Mathematics Department, is for the university to commmit to longer-term funds to cover the teaching of these courses. Note that this is not a question (per se) of how much funds--rather it is a change in the management of the funds. Such a change could have several salutary effects. For example, the Math Department could hire Teaching Post-Docs (TPDs) on a three-year term. This would create a larger pool of potential teachers for Business Math, and create better incentives for teaching performance than the current system. The Department could use this to re-visit the class size question. With instructors who are more committed to the class, can the class size be increased from the current 28 to 32 or 36?

The recognition of the need to integrate the teaching needs and the intellectual objectives of the Math Department also suggests that there is a need for a tenured faculty member to serve as the course coordinator. This individual would ensure continuity across the generations of TPDs. Ideally (s)he would have research interests in common with faculty in the Business School. Unless this bridge between the research and teaching missions of the Department is built, the classes will eventually fall into disrepair.

Another adverse consequence of the current situation is that it makes it harder for students to register for the classes they need. In Fall 2006 for example, some 1,000 students attempted to register for 124 seats in MATH 115b. Of course, as always happens, in time the needed sections are eventually opened. This leaves a bad taste in students' mouths, as registration is prolonged and uncertainties are introduced. Were we of the mind that we had to compete for our students, such a situation would be unacceptable.

It is important to note that the deterioration has not been in the Department's commitment as much as in the Department's ability to act on this and its other commitments. The in-sequence Business Math course has the highest scheduling priority among lower division courses, above calculus and College Algebra. In every previous semester, the Department has begun the term with seats available in the proper Math 115 class when it has run out in other courses. It is true that tenure-track faculty assigned to teach either of these courses has dropped. This is primarily because the retirement of the course’s developer, but it is also part of the general move of tenure-track faculty away from all lower division courses. The rapid increase of Mathematics majors and the decrease in teaching time available from the faculty have forced tenure-track teachers into the higher division.

The Department’s commitment to business math has not waned, but the course has turned out to be much more of a problem than expected. The rather small increases in its budget Math received to start the program were lost almost immediately in the first round of budget rescissions. The Department had only limited success hiring new instructors with the necessary mathematical and computer skills at the $30,000 adjunct faculty level. Math compensated for this by making the course part of the Teaching Post-Doc program that itself only barely survived being eliminated by later cuts. Finally, the academic and computer skills and the maturity level of pre-business majors has not been as high as originally expected, and this make teaching the course at the correct level that more difficult. Of course, math’s role is to act as part of the sieve that separates pre-business students from business students, but that means that Math deals with a good number of students who are not suited for the major they have chosen.

It may be that the solution is not a question of funds, but a question of the management of the funds. Still the Eller College is only one of the University constituents with a major stake in the future of the Mathematics Department. The Department needs outside help in deciding what exactly are the pedagogical needs of the University and which of those needs have the highest priorities when hard decisions are necessary. It does not look like the Math Department will be growing in the near future but it does look like its role in the University is growing rapidly. In the very short term, more efficient money management is absolutely necessary; but soon, something more drastic is just as absolutely necessary.